Obstructions to partitions of chordal graphs

Matrix partition problems generalize graph colouring and homomorphism problems, and occur frequently in the study of perfect graphs. It is difficult to decide, even for a small matrix M, whether the M-partition problem is polynomial time solvable or NP-complete (or possibly neither), and whether M-partitionable graphs can be characterized by a finite set of forbidden induced subgraphs (or perhaps by some other first order condition). We discuss these problems for the class of chordal graphs. In particular, we classify all small matrices M according to whether M-partitionable graphs have finitely or infinitely many minimal chordal obstructions (for all matrices of size less than four), and whether they admit a polynomial time recognition algorithm or are NP-complete (for all matrices of size less than five). We also suggest questions about larger matrices.